The Stacks Project


Tag 0062

Chapter 5: Topology > Section 5.19: Specialization

Lemma 5.19.2. Let $X$ be a topological space.

  1. Any closed subset of $X$ is stable under specialization.
  2. Any open subset of $X$ is stable under generalization.
  3. A subset $T \subset X$ is stable under specialization if and only if the complement $T^c$ is stable under generalization.

Proof. Omitted. $\square$

    The code snippet corresponding to this tag is a part of the file topology.tex and is located in lines 3230–3240 (see updates for more information).

    \begin{lemma}
    \label{lemma-open-closed-specialization}
    Let $X$ be a topological space.
    \begin{enumerate}
    \item Any closed subset of $X$ is stable under specialization.
    \item Any open subset of $X$ is stable under generalization.
    \item A subset $T \subset X$ is stable under specialization
    if and only if
    the complement $T^c$ is stable under generalization.
    \end{enumerate}
    \end{lemma}
    
    \begin{proof}
    Omitted.
    \end{proof}

    Comments (0)

    There are no comments yet for this tag.

    Add a comment on tag 0062

    Your email address will not be published. Required fields are marked.

    In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the lower-right corner).

    All contributions are licensed under the GNU Free Documentation License.




    In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following box. So in case this where tag 0321 you just have to write 0321. Beware of the difference between the letter 'O' and the digit 0.

    This captcha seems more appropriate than the usual illegible gibberish, right?